Galois Theory without abstract algebra

TL;DR

This paper presents an elementary approach to Galois theory using basic polynomial and group algebra, making the subject accessible and providing practical methods for solving polynomials and determining radical solvability.

Contribution

It introduces a Galois theory framework without abstract algebra, including methods for computing Galois groups and radical solutions, with illustrative examples.

Findings

Solution in radicals for lower degree polynomials

Insolubility of the general quintic in radicals

General radical solutions for polynomials when they exist

Abstract

Galois theory is developed using elementary polynomial and group algebra. The method follows closely the original prescription of Galois, and has the benefit of making the theory accessible to a wide audience. The theory is illustrated by a solution in radicals of lower degree polynomials, and the standard result of the insolubility in radicals of the general quintic and above. This is augmented by the presentation of a general solution in radicals for all polynomials when such exist, and illustrated with specific cases. A method for computing the Galois group and establishing whether a radical solution exists is also presented.